inequalities in polar form

Niek Sprakel shared this question 8 months ago
Answered

hi.

Is there any way to create inequalities in polar form in GeoGebra?

For instance, if we have a function y = sin(x), we get a curve, but if

we change it to y < sin(x), we get the area under the curve.

Similarly, we can specify a function in polar form r = sin(t) (where

the radius r is specified as a function of the angle t) and we'd

expect to obtain the area enclosed by the curve by changing it

to r < sin(t).

But I don't see any way to do that with the parametric polar curve command.

Curve((sin(t);t),t,0,2 pi)


Greetings and thanks in advance for any feedback, Niek

Comments (15)

photo
1

Try


Curve((sin(θ); θ), θ, 0, π)
then you can increase its opacity (and also check "inverse filling" if you want)

photo
1

Ah yes, that works, but would it also be possible to obtain logical combinations if you have multiple polar parametric curves?

Similar to how you can do something like (where c is a logical combination of a and b):

a: y < sin(x)

b: x < sin(y)

c: a(x,y) ∧ b(x,y)

photo
1

please, give a concrete example with ρ and θ

like Curve((cos(θ) + sin(θ) - abs(cos(θ) - sin(θ)) / 2; θ), θ, 0, 2π) for ρ<cos(θ)&&ρ<sin(θ)

photo
1

In this example (in the attached ggb file) you can see c1 is easy to define logically in terms of a1 and b1.

If I try something similar for parametric polar curves to obtain the same shapes, I don't see an easy way to obtain c2 from a2 and b2, but perhaps the curves in the example should have the same start and end values for the parameter for this approach to work.

photo
1

Exemple

photo
1

Ah ok, that would be a way, but it doesn't seem feasible for more complicated shapes that intersect in many places.

photo
1

Please give a concrete example

photo
1

Well, for instance, suppose I have a pattern like this and I want something similar in polar form instead of rectangular form?

https://i.imgur.com/jTTAeFT.png

https://i.imgur.com/ulq69S8.png

photo
1

then curve is enough

f(x) must be periodic

Files: foro.ggb
photo
1

No, because I might want to work constructing such a pattern from constituent shapes like rectangles and circles. The logic operators allow for an easy approach to compose complex shapes from basic shapes defined by inequalities.

Why would it only be useful to use these logical operators on inequalities in a rectangular framework and not in a polar framework?

photo
1

6ec9ed1d5d49a2f928a478bda27bb509Ou remplir la forme

photo
1

That's a different visual pattern and I don't see an easy way to come up with a formula for a complicated curve that matches up with the visual pattern obtained from logical combinations of circles and rectangles of varying sizes.

Also, with filling up you have just one option, to fill it up or not (or perhaps two options, since we can invert the filling). With logical combinations you have way more options because different ways of combining logical operators can be used to fill up areas in different ways.

I'm exploring the possibilities of GeoGebra as a tool for creating abstract geometric art along these lines (images found online): https://imgur.com/a/HooEmiW

photo
1

define circles, rectangles, triangles, ovoides o cualquier forma a partir de los datos; crea una herramienta para ello y usala para crear combinaciones de ellas

Files: foro.ggb
photo
1

Oh, I see there is a way to kind of work around it that doesn't involve curves.

photo
photo
1

desigualdad en coordenadas polares deberia ser más especifica

ejemplo: ¿ ρ ≤ θ debería colorear el (1,1)=(sqrt(2);π/4)=(sqrt((2);9π/4) sí o no?

para zonas encerradas por curvas en polares puede ver esto

https://www.geogebra.org/m/...

© 2020 International GeoGebra Institute